Brownian motion: Modeling random fields over arbitrary geometries (stopthrowingrocks.github.io)

This article develops a principled, constructive way to model continuous random fields (functions from a metric space to R) by generalizing the intuition of Brownian motion. Starting from the basic Brownian conditional density P(φ(t1)|φ(t0)) ∝ exp[−(φ1−φ0)^2/(2ν|t1−t0|)], the author derives joint correlation functions for multiple points via the Markov property and shows these joint kernels factorize as products of pairwise conditionals. The exponent is a quadratic form in the field values (sums of (φi−φj)^2/|ti−tj| terms), which both yields an explicit test for whether a generator satisfies the desiderata and suggests a sequential sampling algorithm (sample φ at new points by conditioning on nearby sampled points). A key formal result is that the correlation function is invariant to which variable is treated as “given” — a consequence of the quadratic exponential form and constraints on allowable proportionality constants. Significance for AI/ML and physics: the work supplies an explicit, mathematically tractable kernel-like object for constructing and validating spatial priors and random-field samplers on arbitrary geometries (metric spaces), with direct relevance to Gaussian process modeling, mesh/graph-based spatial simulation, and physics-inspired priors. The paper is technical and symbolic, but its main payoff is an algorithmically useful, provably consistent correlation formula and sampling procedure for continuous random fields.